How do I change the orientation/anchor point of a group?
In edit mode .. not object mode ... move your mesh relative to the origin for an improvement.

or 

in object mode use the menus in the image below.  Object/Transform



Origin to Geometry

For precise control click on part of your mesh to move the 3D Cursor then use the menu option

Origin to 3D Cursor.

Try the similar menu options.

Also I believe you are talking about a Blender Mesh Object and Not a Blender Group.